They wouldn't last long enough to elect a President. This isn't 1850 where nobody relied on electricity, running water, and USD . If the CSA tried to secede again, their state economies would go belly up within seconds and the international community would not recognize them. There wouldn't even need to be a war.
You're right. The US relies on fossil fuels and Texas is #1 in crude production, natural gas production, and electricity production. We have our own electricity interconnection and electricity reliability council.
The US would have plenty of coal left, though.